Almost coconvex approximation of continuous periodic functions

Abstract

If a $2\pi$ -periodic function $f$ continuous on the real axis changes its convexity at $2s, s \in N$, points $y_i : \pi \leq y_{2s} < y_{2s-1} < . . . < y_1 < \pi$ , and, for all other $i \in Z$, $y_i$ are periodically defined, then, for every natural $n \geq N_{y_i}}$, we determine a trigonometric polynomial $P_n$ of order cn such that $P_n$ has the same convexity as $f$ everywhere except, possibly, small neighborhoods of the points $y_i : (y_i \p_i /n, y_i + \pi /n)$, and $\| f P_n\| \leq c(s) \omega 4(f, \pi /n)$,, where $N_{y_i}}$ is a constant depending only on $\mathrm{m}\mathrm{i}\mathrm{n}_{i = 1,...,2s}\{ y_i y_{i+1}\} , c$ and $c(s)$ are constants depending only on $s, \omega 4(f, \cdot )$ is the fourth modulus of smoothness of the function $f$, and $\| \cdot \|$ is the max-norm.